3-Manifolds| 3-Manifolds |
In General
* Topological classification:
A full one has not been found yet, but various different decompositions are
possible, the prime-decomposition, the torus decomposition and the Heegard
decomposition.
* Prime decomposition:
Any three-manifold can be decomposed in
an essentially unique way as
3M = R3 # M1 # M2 # M3 # ... ,
where # stands for a connected sum, the initial R3 factor
is present for the (topologically) asymptotically flat case, and each Mi is
a "prime" manifold;
Notice that, for non-orientable Mi's,
the connected sum # does not specify in which of the two inequivalent ways
the operation is
performed
(for the orientable ones, there is only one orientation-preserving
possibility);
The irreducible pieces Mi have
not been classified (from Rafael,
03.1985).
* Cobordism: All closed 3-manifolds are in the same cobordism class.
* Differentiable structure:
All
closed 3-manifolds have a unique differentiable
structure.
* Spin structure: All 3-manifolds admit a spin structure.
* Decidability: The set of compact 3-manifolds is algorithmically decidable,
i.e., it has an algorithmic description.
Examples > s.a. laplace operator
* Lens spaces: 3D constant
positive curvature manifolds, obtained as
quotients of the three-sphere; Denoted by Lp,q:=
S3/
,
where S3= {(z1,
z2) 
C2 |
(z1,
z2) =
1}, (z1,
z2) (
z1, q z2),
:= exp{2
i/p},
and p and q are
relatively
prime integers; S3 is a p:1 cover; Example:
L2,1 = SO(3).
@ Lens spaces: Bellon CQG(06) [harmonics
from holonomies, and cosmology].
@ Non-orientable: Amendola & Martelli T&A(03), T&A(05)
[small
complexity]; Casali T&A(04) [complexity].
@ Compact hyperbolic: Culler et al Top(98) [smallest]; Kramer ap/04 [group
actions and symmetries].
@ Other: in Freed & Gompf PRL(91)
[Brieskorn sphere]; Scannell CQG(01)
[spacelike
slices of flat spacetimes]; Bray & Neves AM(04)
[prime, Yamabe invariant > RP3];
Boileau & Weidmann Top(05)
[with 2-generated fundamental
group].
Invariants > s.a. knot invariants; Turaev-Viro
Theory.
* Complexity: The numbers
of manifolds of complexity 0, 1, 2, 3, 4, 5, 6, respectively, is 3, 2, 4,
7, 14, 31, 74;
The
first hyperbolic 3-manifold occurs at complexity 9.
* Volume: For hyperbolic
ones,
with curvature normalized to –1; Problem: For each V, there
are finitely many 3-manifolds with volume < V;
Which is the smallest?
* Other: Reshetikhin-Turaev
invariant; > s.a.
topological field theories, including chern-simons, spin
networks.
@ General references: Stewart Nat(89)mar
[volume]; Reshetikhin & Turaev IM(91)
[from links and
quantum
groups];
Kauffman & Lins 94; Bott & Cattaneo JDG(98)dg/97,
JDG(99)m.GT/98
[integral];
Liu Top(99);
Korepanov JNMP(01)m.GT/00 [PL];
Korepanov & Martyushev JNMP(02);
Turaev 02 [torsion]; McDuff BAMS(06)
[and Floer theory, Ozsváth-Szabó]; King T&A(07)
[ideal Turaev-Viro
invariant].
@ And topological quantum field theory: Bakalarska & Broda ht/99-in,
FdP(00)ht/99;
Ramadevi & Naik CMP(00)
[Lickorish
invariant and Chern-Simons theory]; Kaul & Ramadevi CMP(01)ht/00 [from
Chern-Simons theory]; Garnerone et al qp/07 [in
SU(2) Chern-Simons-Witten topological
quantum field theory].
@ Finite-type: Garoufalidis et al G&T(01)m.GT/00 [and
trivalent graphs].
@ Relationships: Guadagnini & Pilo CMP(98); Mariño & Moore
NPB(99) [and 4D Donaldson-Witten invariants].
@ With boundary: Murakami & Ohtsuki CMP(97) [from universal quantum
invariant].
@ Classification: Milnor AJM(62); Hendricks
BAMS(77);
Thompson BAMS(98)
[algorithmic]; Morgan BAMS(05).
@ Related topics: Baseilhac & Benedetti m.GT/01;
Harvey
G&T(02)
[cut number not bounded below by 
1(X)/3],
Top(05)
[from fundamental group]; Ozsváth & Szabó AM(04)
[and holomorphic disks]; Cavicchioli & Spaggiari DM(08) [genus-2, representation
by
family
of integers].
Other Structure and Concepts > s.a. diffeomorphisms [including
Smale conjecture];
embeddings; 3D
geometry; knots.
@ General references: Gluck & Pan Top(98)
[embedded 2-surfaces]; Neumann & Swarup G&T(97)
[decompositions]; Camacho & Camacho T&A(07)
[codimension-1 foliations]; Fernández & Mira DG&A(07)
[constant mean curvature
surfaces in homogeneous 3-manifolds].
@ Hyperbolic: Fenley Top(98) [foliations]; Gabai et al AM(03) [homotopy
hyperbolic].
References > s.a. conjectures [Smith]; topological
field theory.
@ Simple, or for physicists: Thurston & Weeks SA(84)jul; Giulini
IJTP(94)gq/93; Thurston CQG(98).
@ Topology: Neuwirth 75; Hempel 76; Thurston 78; Jaco 80; Brown & Thickstun
ed-82; Bing 83; Fenn ed-85; Thurston 97.
@ And computers: Lins 95 [gems].
@ Homeomorphisms: Cesar de Sa & Rourke BAMS(79).
@ Surgery / Links: in Rolfsen 76, ch9; Kirby IM(78).
@ Framings: Atiyah Top(90).
@ Related topics: Schoen & Yau PNAS(78), AM(79), PRL(79) & refs;
Hendriks & Laudenbach
Top(84);
Friedman & Witt Top(86);
Freedman & Feng
89; Gabai & Oertel AM(89); Crane CMP(91).
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jul 2008